The combination of strong spin-orbit coupling and ultra-low symmetry gives rise to many unique properties in materials.
One of the most curious is
nonreciprocal directional dichroism or `one-way transparency'.[1]}, [2]}
Spectroscopically, the effect
occurs near a magnetoelectric excitation, often in the THz[3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]} but sometimes extending to the optical
region and beyond,[14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]} and it arises from the fact that counter-propagating beams can have different absorption coefficients.
[21]}, [1]}, [23]}, [2]}, [25]}
In other words, a
sample may be highly transmitting when measured with light
in the +k direction but nearly opaque in the
-k direction.
Nonreciprocity has a number of symmetry prerequisites. [23]}, [25]} For a given light propagation direction \(\vec{\textrm {k}}\), all
symmetries that reverse +k to -k
while leaving the sample unchanged
must be broken including
inversion, mirrors or \(C_2\)  rotations about a plane or axis
perpendicular to \(\vec{\textrm {k}}\), and time-reversal.[21]}
Several measurement configurations can be used to
take advantage of various types of symmetry breaking.[23]}, [25]} Toroidal
dichroism occurs when light propagation
is along the toroidal moment T (k \(\parallel \)  T = P
\(\times \)  M, where P and M are the electric polarization and
magnetic moment).[9]}, [5]}, [33]}, [34]}, [10]}, [36]}
Magnetochiral dichroism, on the other hand, involves in both chirality and applied magnetic field.[8]}, [15]}, [16]}, [40]}, [41]}, [42]}, [43]}, [44]}, [45]}, [46]}
Interestingly, nonreciprocity can be realized by reversing the external magnetic
field as well as the light propagation direction. Linearly- and circularly-polarized light offer additional degrees of freedom,[10]} and with a vortex beam, one can even select specific projections of the orbital angular momentum.[12]}, [13]} The spectroscopic response of these eigenstates is not always equal and opposite, leading to the phenomenon of nonreciprocal directional anisotropy. As a result, unpolarized light can reveal these effects as well. [16]}, [6]}, [20]}
